Dirichlet series
| L(s) = 1 | − 5.90e20·4-s − 1.50e29·7-s + 2.78e38·13-s + 3.48e41·16-s − 2.61e44·19-s − 1.69e48·25-s + 8.86e49·28-s − 1.23e51·31-s + 1.44e54·37-s + 4.34e56·43-s + 2.07e57·49-s − 1.64e59·52-s + 7.51e61·61-s − 2.05e62·64-s + 1.08e63·67-s + 2.88e64·73-s + 1.54e65·76-s − 2.79e64·79-s − 4.18e67·91-s − 6.35e68·97-s + 9.99e68·100-s + 4.12e69·103-s + 1.28e70·109-s − 5.23e70·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.04·7-s + 1.03·13-s + 16-s − 1.99·19-s − 25-s + 1.04·28-s − 0.435·31-s + 1.13·37-s + 1.91·43-s + 0.101·49-s − 1.03·52-s + 1.91·61-s − 64-s + 1.08·67-s + 1.50·73-s + 1.99·76-s − 0.0952·79-s − 1.08·91-s − 1.81·97-s + 100-s + 1.48·103-s + 0.659·109-s − 1.04·112-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(70-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+69/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(9\) = \(3^{2}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(271.363\) |
| Root analytic conductor: | \(16.4731\) |
| Motivic weight: | \(69\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | yes |
| Self-dual: | yes |
| Analytic rank: | \(1\) |
| Selberg data: | \((2,\ 9,\ (\ :69/2),\ -1)\) |
Particular Values
| \(L(35)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{71}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{69} T^{2} \) |
| 5 | \( 1 + p^{69} T^{2} \) | |
| 7 | \( 1 + \)\(15\!\cdots\!20\)\( T + p^{69} T^{2} \) | |
| 11 | \( 1 + p^{69} T^{2} \) | |
| 13 | \( 1 - \)\(27\!\cdots\!30\)\( T + p^{69} T^{2} \) | |
| 17 | \( 1 + p^{69} T^{2} \) | |
| 19 | \( 1 + \)\(26\!\cdots\!96\)\( T + p^{69} T^{2} \) | |
| 23 | \( 1 + p^{69} T^{2} \) | |
| 29 | \( 1 + p^{69} T^{2} \) | |
| 31 | \( 1 + \)\(12\!\cdots\!72\)\( T + p^{69} T^{2} \) | |
| 37 | \( 1 - \)\(14\!\cdots\!90\)\( T + p^{69} T^{2} \) | |
| 41 | \( 1 + p^{69} T^{2} \) | |
| 43 | \( 1 - \)\(43\!\cdots\!80\)\( T + p^{69} T^{2} \) | |
| 47 | \( 1 + p^{69} T^{2} \) | |
| 53 | \( 1 + p^{69} T^{2} \) | |
| 59 | \( 1 + p^{69} T^{2} \) | |
| 61 | \( 1 - \)\(75\!\cdots\!42\)\( T + p^{69} T^{2} \) | |
| 67 | \( 1 - \)\(10\!\cdots\!80\)\( T + p^{69} T^{2} \) | |
| 71 | \( 1 + p^{69} T^{2} \) | |
| 73 | \( 1 - \)\(28\!\cdots\!90\)\( T + p^{69} T^{2} \) | |
| 79 | \( 1 + \)\(27\!\cdots\!24\)\( T + p^{69} T^{2} \) | |
| 83 | \( 1 + p^{69} T^{2} \) | |
| 89 | \( 1 + p^{69} T^{2} \) | |
| 97 | \( 1 + \)\(63\!\cdots\!70\)\( T + p^{69} T^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.883634942166028569237376229809, −8.982365065187612722812362040354, −8.051567365250850791926321424977, −6.50546147971038922229027053182, −5.69675770308784689599518497468, −4.23691778761541126082171461831, −3.64729824198935893221238930643, −2.28990813947985077155636110024, −0.861450780678997611319749000210, 0, 0.861450780678997611319749000210, 2.28990813947985077155636110024, 3.64729824198935893221238930643, 4.23691778761541126082171461831, 5.69675770308784689599518497468, 6.50546147971038922229027053182, 8.051567365250850791926321424977, 8.982365065187612722812362040354, 9.883634942166028569237376229809